## On $$n$$-circled $$\mathcal{H}^ \infty$$-domains of holomorphy.(English)Zbl 0884.32011

A domain $$G\subset\mathbb{C}^n$$ is called $$n$$-circled if for every $$(z_2,\dots,z_n)\in G$$ and any $$(\theta_2,\dots,\theta_n)\in\mathbb{R}^n(e^{i\theta_1}z_1,\dots,e^{i\theta_n}z_n) \in G$$. Let $${\mathcal F}$$ be some subspace in $$H^\infty(G)$$, for example $${\mathcal F}=A^k(G)$$. $$G$$ is said to be an $${\mathcal F}$$-domain of holomorphy if for any pair of domains $$G_0$$, $$\widetilde G\subset\mathbb{C}^n$$ with $$\emptyset\neq G_0\subset\widetilde G\cap G$$, $$\widetilde G\not\subset G$$, there exists a function $$f\in{\mathcal F}$$ such that $$f|G_0$$ is not the restriction of a function $$\widetilde f\in{\mathcal O}(\widetilde G)$$.
The purpose of this paper is to present various characterizations of $$n$$-circled domains of holomorphy with respect to some distinguished subspaces of $$H^\infty(G)$$.

### MSC:

 32D05 Domains of holomorphy

### Keywords:

$$n$$-circled domains of holomorphy
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